Quantum (dual) Grassmann superalgebra as $mathcal U_q(mathfrak{gl}(m|n))$-module algebra and beyond

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $textrm{Diff}_q(Omega_q)$ defined over $Omega_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Omega_q(m|n)$. An interested point of this approach here is that even though $mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $Omega_q$ is made into the $mathcal U_q(mathfrak g)$-module (super)algebra structure,$Omega_q=Omega_q(m|n)$ for $q$ generic, or $Omega_q(m|n, bold 1)$ for $q$ root of unity, and $mathfrak g=mathfrak{gl}(m|n)$ or $mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $mathcal U_q(mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $Omega_q^!$ as $mathcal U_q(mathfrak g)$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in cite{Ma}, & cite{Ma1}.